{"title":"Spectral characterization of the constant sign Green's functions for periodic and Neumann boundary value problems of even order","authors":"A. Cabada, Lucía López-Somoza","doi":"10.7153/dea-2022-14-24","DOIUrl":null,"url":null,"abstract":". In this paper we will characterize the interval of real parameters M in which the Green’s function G M , related to the operator T 2 n [ M ] u ( t ) : = u ( 2 n ) ( t )+ Mu ( t ) coupled to periodic, u ( i ) ( 0 ) = u ( i ) ( T ) , i = 0 ,..., 2 n − 1, or Neumann, u ( 2 i + 1 ) ( 0 ) = u ( 2 i + 1 ) ( T ) = 0, i = 0 ,..., n − 1, boundary conditions, has constant sign on its square of de fi nition. More concisely, we will prove that the optimal values are given as the fi rst zeros of G M ( 0 , 0 ) or G M ( T / 2 , 0 ) , depending both on the sign of G M and on the fact whether 2 n is, or is not, a multiple of 4. Such values will be characterized as the eigenvalues of the operator u ( 2 n ) related to suitable boundary conditions. This characterization allows us to obtain such values without calculating the exact expression of the Green’s function.","PeriodicalId":179999,"journal":{"name":"Differential Equations & Applications","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/dea-2022-14-24","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
. In this paper we will characterize the interval of real parameters M in which the Green’s function G M , related to the operator T 2 n [ M ] u ( t ) : = u ( 2 n ) ( t )+ Mu ( t ) coupled to periodic, u ( i ) ( 0 ) = u ( i ) ( T ) , i = 0 ,..., 2 n − 1, or Neumann, u ( 2 i + 1 ) ( 0 ) = u ( 2 i + 1 ) ( T ) = 0, i = 0 ,..., n − 1, boundary conditions, has constant sign on its square of de fi nition. More concisely, we will prove that the optimal values are given as the fi rst zeros of G M ( 0 , 0 ) or G M ( T / 2 , 0 ) , depending both on the sign of G M and on the fact whether 2 n is, or is not, a multiple of 4. Such values will be characterized as the eigenvalues of the operator u ( 2 n ) related to suitable boundary conditions. This characterization allows us to obtain such values without calculating the exact expression of the Green’s function.